In order to avoid the heavy computational cost of nuclear norm minimization for all considered convex models (besides “linear", which is just a logistic regression with standard weight decay), we rely on the Burer-Monteiro factorization [1]}. In particular, if one has the problem
\(\min _{\mathbf {Z} \in \mathbb {R}^{a \times c}} \sum _{i=1}^n \mathcal {L}(f_i(\mathbf {Z}), \mathbf {Y}_i) + \beta \Vert \mathbf {Z}\Vert _*,\) 
